Conjecture 1 ([1]}, [2]}) 
Let \(a_1,\ldots , a_\ell \colon [c,+\infty )\rightarrow {\mathbb {R}}\)  be functions from a Hardy field of at most polynomial growth such that every non-trivial linear combination of these functions
stays logarithmically away from rational polynomials. Then the sequences \([a_1(n)],\ldots , [a_\ell (n)]\)  are jointly ergodic.

